Jelavic's goal, an excellent volley from a Leon Osman lay-off in the first half, was enough to inflict a third straight defeat on Spurs.
Arsenal could now move to within a point of them with victory over Newcastle on Monday night, while Everton are now 10 games unbeaten in all competitions and head into Tuesday's Merseyside derby at Anfield just two points behind Liverpool.
The last Spurs manager to preside over three straight defeats was Juande Ramos in October 2008 and it resulted in him being replaced by Harry Redknapp.
For all the talent the visitors had on display it was Jelavic who stole the headlines as he opened his account just 77 minutes into his Everton career, having made just two previous substitute appearances.
When the 26-year-old arrived at Goodison Park in a £5.5million transfer from Rangers in January he was asked what he would bring to the club.
"Goals, goals, goals" was the response and he started to deliver on that promise with a sumptuous strike midway through the first half.
Jelavic had looked a threat from the start, generally offering a willing outlet up front, and moments before his goal he provided the knockdown from which Marouane Fellaini forced Brad Friedel to save an angled shot with his legs.
In the 22nd minute he played a pivotal part with the sort of finish Everton have been missing since the sale of Louis Saha in January.
The Frenchman was watching from the bench, although David Moyes' claims he had a gentlemen's agreement with Redknapp was disproved in the second half when the striker was sent on.
Leon Osman broke past Younes Kaboul into the box and brilliantly took both Sandro and Ledley King out of the game with his cutback which Jelavic clinically fired first-time past Friedel.
Jelavic did not appear to be lacking in confidence and with established free-kick takers Leighton Baines and Drenthe standing over the ball he whipped in a right-footed shot which was turned away by the diving Spurs goalkeeper.
Most of the threat before the match had been expected to come from Drenthe and Tottenham winger Gareth Bale.
Redknapp's decision to post the Wales international on the right meant Drenthe had to be switched to the opposite flank in order for Seamus Coleman to track Bale.
It also meant Moyes could keep a closer eye on the Dutchman who excites and infuriates in equal measure with his unpredictability.
Throughout most of the first half the Everton manager instructed, cajoled and ordered the on-loan Real Madrid midfielder into where he wanted him.
It seemed to work and only when Bale briefly appeared on his usual left side shortly before the interval did Drenthe display his less disciplined persona when well out of ear-shot of Moyes.
Spurs were disappointing in the first half, Bale particularly so with his main contribution being two wide shots.
Jermain Defoe was also guilty of wastefulness as he seemed destined to score until some late pressure from John Heitinga forced him to shoot well wide.
The striker looked sharper immediately after the restart as Howard twice had to save from him, the second after Sylvain Distin had backed off and given him far too much room inside the penalty area.
Luka Modric was also starting to look more threatening and he too was denied by a Distin block before also firing over.
Saha arrived in the 54th minute for the ineffectual Emmanuel Adebayor but Defoe looked Tottenham's greatest threat and his deflected shot briefly caused Howard a moment of concern before he twisted his body mid-save.
While Tottenham's pressure was growing the home side were still a threat and when Coleman burst from the halfway line to tee up Osman Friedel had to save again.
Redknapp's decision to persist with Bale on the right was not welcomed by the visiting fans, who sang "Gareth Bale, he plays on the left" as the game entered its final quarter.
From his right-sided position the Welshman thought he had set up the equalising goal when he turned Benoit Assou-Ekotto's cross into the path of Defoe but the striker was well offside as he fired in from close range.
Saha should have scored it in the third minute of added time but hit a post.
It would have been the cruellest of finishes.